Transcript Appendix C -- A Refresher on Probability and Statistics

A Refresher on Probability and Statistics
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What We’ll Do ...
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
Ground-up review of probability and statistics necessary
to do and understand simulation
Outline
—
Probability – basic ideas, terminology
—
Random variables, joint distributions
—
Sampling
—
Statistical inference – point estimation, confidence
intervals, hypothesis testing
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Monte Carlo Simulation
Monte Carlo method: Probabilistic simulation technique used when a
process has a random component

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Identify a probability
distribution
Setup intervals of
random numbers to
match probability distribution
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Obtain the random numbers
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Interpret the results
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Probability Basics
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Experiment – activity with uncertain outcome
—
Flip coins, throw dice, pick cards, draw balls from urn, …
—
Drive to work tomorrow – Time? Accident?
—
Operate a (real) call center – Number of calls? Average
customer hold time? Number of customers getting busy
signal?
—
Simulate a call center – same questions as above
Sample space – complete list of all possible individual
outcomes of an experiment
—
Could be easy or hard to characterize
—
May not be necessary to characterize
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Probability Basics (cont’d.)
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Event – a subset of the sample space
—
Describe by either listing outcomes, “physical” description, or
mathematical description
—
Usually denote by E, F, G… or E1, E2, etc.
—
Ex: arrival of a customer, start of work on a job
Probability of an event is the relative likelihood that it will occur when
you do the experiment
—
A real number between 0 and 1 (inclusively)
—
Denote by P(E), P(E  F), etc.
—
Interpretation – proportion of time the event occurs in many
independent repetitions (replications) of the experiment
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Probability Basics (cont’d.)
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Some properties of probabilities
If S is the sample space, then P(S) = 1
E
S
F
If Ø is the empty event (empty set), then P(Ø) = 0
If EC is the complement of E, then P(EC) = 1 – P(E)
P(E  F) = P(E) + P(F) – P(E  F)
If E and F are mutually exclusive (i.e., E  F = Ø), then
P(E  F) = P(E) + P(F)
If E is a subset of F (i.e., the occurrence of E implies the occurrence of F),
then P(E)  P(F)
If o1, o2, … are the individual outcomes in the sample space, then
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Probability Basics (cont’d.)
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Conditional probability
—
Knowing that an event F occurred might affect the
probability that another event E also occurred
—
Reduce the effective sample space from S to F, then
measure “size” of E relative to its overlap (if any) in F,
rather than relative to S
—
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Definition (assuming P(F)  0):
E and F are independent if P(E  F) = P(E) P(F)
—
Implies P(E|F) = P(E) and P(F|E) = P(F), i.e., knowing
that one event occurs tells you nothing about the other
—
If E and F are mutually exclusive, are they independent?
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Random Variables
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One way of quantifying, simplifying events and
probabilities
A random variable (RV) is a number whose value is
determined by the outcome of an experiment
—
Assigns value to each point in the sample space
—
Associates with each possible outcome of the
experiment
—
Usually denoted as capital letters: X, Y, W1, W2, etc.
Probabilistic behavior described by distribution function
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Discrete vs. Continuous RVs
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Two basic “flavors” of RVs, used to represent or model
different things
Discrete – can take on only certain separated values
—
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Number of possible values could be finite or infinite
Continuous – can take on any real value in some range
—
Number of possible values is always infinite
—
Range could be bounded on both sides, just one side, or
neither ( ∞    ∞ )
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RV in Simulation
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Input
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Uncertain time duration (service or inter-arrival times)
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Number of customers in an arriving group
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Which of several part types a given arriving part is
Output
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Average time in system
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Number of customers served
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Maximum length of buffer
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Discrete Distributions
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Let X be a discrete RV with possible values (range) x1, x2,
… (finite or infinite list)
Probability Mass Function (PMF)
p(xi) = P(X = xi) for i = 1, 2, ...
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The statement “X = xi” is an event that may or may not
happen, so it has a probability of happening, as
measured by the PMF
—
Can express PMF as numerical list, table, graph, or
formula
—
Since X must be equal to some xi, and since the xi’s are
all distinct,
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Discrete Distributions (cont’d.)
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Cumulative distribution function (CDF) – probability that
the RV will be  a fixed value x:
1.0
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Properties of discrete CDFs
0.5
0  F(x)  1 for all x
As x  – , F(x)  0
As x  + , F(x)  1
F(3)
F(2)
F(1)
F(x) is nondecreasing in x
F(x) is a step function continuous from the right with
jumps at the xi’s of height equal to the PMF at that xi12
Example of CDF
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Example of CDF
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Discrete Distributions (cont’d.)
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Computing probabilities about a discrete RV – usually use
the PMF
—
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Add up p(xi) for those xi’s satisfying the condition for
the event
With discrete RVs, must be careful about weak vs. strong
inequalities – endpoints matter!
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Discrete Expected Values
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Data set has a “center” – the average (mean)
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RVs have a “center” – expected value
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Also called the mean or expectation of the RV X
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Other common notation: m, mX
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Weighted average of the possible values xi, with weights being their
probability (relative likelihood) of occurring
—
What expectation is not: The value of X you “expect” to get
E(X) might not even be among the possible values x1, x2, …
—
What expectation is:
Repeat “the experiment” many times, observe many X1, X2, …, Xn
E(X) is what
converges to (in a certain sense) as n  
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Discrete Variances and
Standard Deviations
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Data set has measures of “dispersion” –
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Sample variance
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Sample standard deviation
RVs have corresponding measures
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Other common notation:
—
Weighted average of squared deviations of the possible values xi
from the mean
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Standard deviation of X is
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Interpretation analogous to that for E(X)
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Continuous Distributions
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Now let X be a continuous RV
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Possibly limited to a range bounded on left or right or
both
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No matter how small the range, the number of possible
values for X is always (uncountably) infinite
—
Not sensible to ask about P(X = x) even if x is in the
possible range
—
Technically, P(X = x) is always 0
—
Instead, describe behavior of X in terms of its falling
between two values
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Continuous Distributions (cont’d.)
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Probability density function (PDF) is a function f(x) with
the following three properties:
f(x)  0 for all real values x
The total area under f(x) is 1:
For any fixed a and b with a  b, the probability that X
will fall between a and b is the area under f(x) between
a and b:
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CDF and PDF
P = P[ X  x] = Probability that the
random variable X is less than or
equal to a given number x
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Continuous Distributions (cont’d.)
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Cumulative distribution function (CDF) - probability that the RV
will be  a fixed value x:
F(x) may or
may not have
a closed-form
formula
Properties of continuous CDFs
0  F(x)  1 for all x
As x  –, F(x)  0
As x  +, F(x)  1
These four properties
are same as
discrete CDFs
F(x) is nondecreasing in x
F(x) is a continuous function with slope equal to the PDF:
f(x) = F'(x)
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Continuous Expected Values, Variances, and Standard
Deviations
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Expectation or mean of X is
—
Roughly, a weighted “continuous” average of possible
values for X
—
Same interpretation as in discrete case: average of a
large number (infinite) of observations on the RV X
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Variance of X is
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Standard deviation of X is
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Joint Distributions
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So far: Looked at only one RV at a time
But they can come up in pairs, triples, …, tuples, forming
jointly distributed RVs or random vectors
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Input: (T, P, S) = (type of part, priority, service time)
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Output: {W1, W2, W3, …} = output process of times in
system of exiting parts
One central issue is whether the individual RVs are
independent of each other or related
Will take the special case of a pair of RVs (X1, X2)
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Extends naturally (but messily) to higher dimensions
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Joint Distributions (cont’d.)
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Joint CDF of (X1, X2) is a function of two variables
Replace
“and” with “,”
—
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Same definition for discrete and continuous
If both RVs are discrete, define the joint PMF
If both RVs are continuous, define the joint PDF f(x1, x2) as a
nonnegative function with total volume below it equal to 1,
and
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Covariance Between RVs
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Measures linear relation between X1 and X2
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Covariance between X1 and X2 is
—
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Covariance tells us whether the two random variables are
related or not. If they are, whether the relationship is
positive or negative.
Interpreting value of covariance – difficult since it depends
on units of measurement
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Correlation Between RVs
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Correlation (coefficient) between X1 and X2 is
—
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Always between –1 and +1
Ex: Correlation of 0.85 means strong relationship, 0.10
means weak.
—
Cor (X, Y) > 0 means +ve Correlation
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X & Y move in the same direction  
—
Cor (X, Y) = 0 means no correlation
—
Cor X, Y) < 0 means –ve correlation X , and Y 
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Independent RVs
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X1 and X2 are independent if their joint CDF factors into the product of
their marginal CDFs:
—
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Equivalent to use PMF or PDF instead of CDF
Properties of independent RVs:
— They have nothing (linearly) to do with each other
— Independence  uncorrelated
—
But not vice versa, unless the RVs have a joint normal distribution
Tempting just to assume it whether justified or not
Independence in simulation
— Input: Usually assume separate inputs are indep. – valid?
— Output: Standard statistics assumes indep. – valid?!?!?!?
—
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Sampling
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Statistical analysis – estimate or infer something about a
population or process based on only a sample from it
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Think of a RV with a distribution governing the
population
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Random sample is a set of independent and identically
distributed (IID) observations X1, X2, …, Xn on this RV
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In simulation, sampling is making some runs of the
model and collecting the output data
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Don’t know parameters of population (or distribution)
and want to estimate them or infer something about
them based on the sample
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Sampling (cont’d.)
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Population parameter
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Sample estimate
Population mean m = E(X)
Sample mean
Population variance s2
Sample variance
Population proportion
Sample proportion
Parameter – need to know whole
population
Fixed (but unknown)
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Sample statistic – can be
computed from a sample
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Varies from one sample to
another – is a RV itself, and has
a distribution, called the
sampling distribution
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Point Estimation
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A sample statistic that estimates (in some sense) a
population parameter
Properties
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Unbiased: E(estimate) = parameter
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Efficient: Var(estimate) is lowest among competing
point estimators
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Consistent: Var(estimate) decreases (usually to 0) as the
sample size increases
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Confidence Intervals
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A point estimator is just a single number, with some uncertainty or
variability associated with it
Confidence interval quantifies the likely imprecision in a point estimator
—
—
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An interval that contains (covers) the unknown population parameter
with specified (high) probability 1 – a
Called a 100 (1 – a)% confidence interval for the parameter
Confidence interval for the population mean m:
tn-1,1-a/2 is point below which is area
1 – a/2 in Student’s t distribution with
n – 1 degrees of freedom
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CIs for some other parameters – in text book
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Confidence Intervals in Simulation
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Run simulations, get results
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View each replication of the simulation as a data point
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Random input  random output
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Form a confidence interval
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Brackets (with probability 1 – a) the “true” expected output
(what you’d get by averaging an infinite number of
replications)
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Example
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1.2, 1.5, 1.68, 1.89, 0.95, 1.49, 1.58, 1.55, 0.50, 1.09.
Calculate the 90% confidence interval
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Sample Mean =
= 1.34
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Sample Variance = s2 = 0.17l
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90% confidence interval means: a = 1 – 0.90 = 0.1
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Degrees of freedom: n = 10 – 1 = 9.
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1.34  t9,0.95  (0.17 / 10). Look into ‘t’ distribution table for t9,0.95 = 1.83
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1.34  1.83  (0.17 / 10). = 1.34  0.24
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 Confidence Interval = [1.10, 1.58]
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Hypothesis Tests
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Test some assertion about the population or its parameters
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Null hypothesis (H0) – what is to be tested
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Alternate hypothesis (H1 or HA) – denial of H0
H0: m = 6
vs. H1: m  6
H0: s < 10 vs. H1: s  10
H0: m1 = m2 vs. H1: m1  m2
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Develop a decision rule to decide on H0 or H1 based on
sample data
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Errors in Hypothesis Testing
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Type-I error is often called the producer's risk
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The probability of a type-I error is the level of significance of the test of hypothesis and is denoted by a .
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Type-II error is often called the consumer's risk for not rejecting possibly a worthless product
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The probability of a type-II error is denoted by b . The quantity 1 - b is known as the Power of a Test
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H0 and H1 are not given equal treatment. Benefit of doubt is given to H0
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p-Values for Hypothesis Tests
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Traditional method is “Accept” or Reject H0
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Alternate method – compute p-value of the test
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p-value = probability of getting a test result more in favor of H1
than what you got from your sample
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Small p (< 0.01) is convincing evidence against H0
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Large p (> 0.10) indicates lack of evidence against H0
Connection to traditional method
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If p < a, reject H0
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If p  a, do not reject H0
p-value quantifies confidence about the decision
A p-value is a measure of how much evidence you have
against the null hypothesis
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Goodness-of-fit Test
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Chi – Square Test
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Kolmogorov – Smirnov test
— Both
tests ask how close the fitted
distribution is to the empirical distribution
defined directly by the data
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Hypothesis Testing in Simulation
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Input side
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Specify input distributions to drive the simulation
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Collect real-world data on corresponding processes
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“Fit” a probability distribution to the observed real-world data
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Test H0: the data are well represented by the fitted distribution
Output side
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Have two or more “competing” designs modeled
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Test H0: all designs perform the same on output, or test H0: one
design is better than another
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Case Study
Case Study: Printed Circuit Assembly
Manufacturing
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The company, engaged in electronic assembly
contract manufacturing, wants to achieve the
following goals:
— Maximize
equipment utilization
— Minimize
machine downtime
— Increase
— Provide
inventory control accuracy
material traceability
— Minimize
time and resources spent looking for
materials and tools on the shop-floor
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Electronics Assembly
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Surface Mount Technology (SMT) or Pin Through-Hole (PTH) are
used to place components on bare boards
An SMT assembly line typically include:
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Screen printer - to apply solder paste on the bare board
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High-speed placement machine - for chips typically
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Fine-Pitch placement machine - for larger components typically
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Owen - to bake the board after components are placed.
The Company has 3 assembly lines
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Typical Reasons for Assembly Line Down
Time
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Poor line balance and flexibility
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Poor machine balance within assembly lines
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Large number of setups and total setup time
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Part shortage during the run
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Feeder problems
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Long reel changeovers
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Operator is not attending the machine
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Setup kit is not delivered on time
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Placing wrong parts
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Component data problems
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Process Control – 1st piece inspection
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Operator waiting for support
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Machine program changeover time
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Real-Time Performance Monitoring
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Machine Utilization
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Machine Utilization
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Assembly Line Performance Metrics
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Assembly efficiency - the difference (in percentage)
between the desired assembly time and the actual
assembly time required to complete a board
(desired time/actual time)*100; target = 95-100%
Minimum cycle time - the largest machine
operation time within the assembly line
Average cycle time - the average time a board is
completed, i.e. the last operation is completed
The average number of boards in the queue between two placement machines
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A Guided Tour Through Arena
Flowchart and Spreadsheet Views
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Model window split into two views
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Flowchart view
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Graphics
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Process flowchart
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Animation, drawing
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Edit things by double-clicking on them, get into a dialog
Spreadsheet view
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Displays model data directly
—
Can edit, add, delete data in spreadsheet view
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Displays all similar kinds of modeling elements at once
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Many model parameters can be edited in either view
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Horizontal splitter bar to apportion the two views
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View/Split Screen to see only the most recently selected view
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Modules
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Basic building blocks of a simulation model
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Two basic types: flowchart and data
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Different types of modules for different actions, specifications
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“Blank” modules are on the Project Bar
—
To add a flowchart module to your model, drag it from the
Project Bar into the flowchart view of the model window
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To use a data module, select it (single-click) in the Project
Bar and edit in the spreadsheet view of the model window
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Relations Among Modules
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Flowchart and data modules are related via names for
objects
—
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Queues, Resources, Entity types, Variables … others
Arena keeps internal lists of different kinds of names
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Presents existing lists to you where appropriate
—
Helps you remember names, protects you from typos
All names you make up in a model must be unique across
the model, even across different types of modules
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Create Module
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Process Module
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Queue-Length Plot
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Dispose Module
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Setting the Run Conditions
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Run/Setup menu dialog – five tabs
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Project Parameters – Title, your name, output statistics
—
Replication Parameters – Number of Replications,
Length of Replication (and Time Units), Base Time
Units (output measures, internal computations),
Warm-up Period (when statistics are cleared),
Terminating Condition (complex stopping rules),
Initialization options Between Replications
—
Other three tabs specify animation speed, run
conditions, and reporting preferences
Terminating your simulation:
 You must specify – part of modeling
 Arena has no default termination
 If you don’t specify termination, Arena will
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usually keep running forever
Viewing the Reports
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Click Yes in the Arena box at the end of the run
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Opens up a new reports window (separate from model
window) inside the Arena window
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Project Bar shows Reports panel, with different reports
(each one would be a new window)
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Remember to close all reports windows before future runs
Default installation shows Category Overview report –
summarizes many things about the run
—
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Reports have “page” to browse Also, “table contents” tree
at left for quick jumps via
Times are in Base Time Units for the model
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Types of Statistics Reported
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Many output statistics are one of three types:
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Tally – avg., max, min of a discrete list of numbers
—
—
Time-persistent – time-average, max, min of a plot of
something where the x-axis is continuous time
—
—
Used for discrete-time output processes like waiting times in
queue, total times in system
Used for continuous-time output processes like queue lengths,
WIP, server-busy functions (for utilizations)
Counter – accumulated sums of something, usually just
nose counts of how many times something happened
—
Often used to count entities passing through a point in the model
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Homework 2
Work as a team of 2.
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Problem 1: Question C4 from Appendix C
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Problem 2: Question 3.6
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Due 9/9/03.
—
Electronic submission
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